应用优化的同伦分析法求解非线性Jerk方程

摘要

An optimal homotopy analysis method was applied to calculate approximate periods and approximately analytical periodic solutions to a third-order differential equation with cubic nonlinearities. An example showed that the second-order approximately analytical periodic solution is easily obtained via optimal homotopy analysis method; when the initial velocity amplitude a is large, the largest percentage error of the first-order approximate period to the exact one is -0.415% , and the largest percentage error of the second-order approximate period is -0.0298%. A comparison of the approximately analytical periodic solutions with the numerical exact ones showed that the first-order and second-order approximately analytical periodic solutions have very high accuracy. It was demonstrates that optimal homotopy analysis method is very effective for solving a nonlinear Jerk equation.%应用优化的同伦分析法计算了具有三次非线性项的三阶微分方程( Jerk)的近似周期和近似解析周期解.给出一个算例说明由优化的同伦分析法可以容易得到精确的二阶近似周期解.当初速度a比较大时,一阶近似周期与精确周期的百分比误差为-0.415％,而二阶近似周期与精确周期的百分比误差为-0.0298％.与数值方法给出的“精确”周期解比较,一阶近似解析周期解和二阶近似周期解的精度很高.这个说明同伦分析法对求解非线性Jerk方程非常有效.